Quadratic Equations

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A quadratic equation is an equation involving one variable with all indices as whole numbers and having 2 as the maximum index of the variable. General form: ax² + bx + c = 0, where a, b, c are real n

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Comparing with ax² + bx + c = 0: a = 3 (coefficient of x²) b = -7 (coefficient of x) c = 2 (constant term)

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The roots of a quadratic equation are the values of the variable that make the equation equal to zero. If x = α is a root of ax² + bx + c = 0, then aα² + bα + c = 0.

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x² - 5x + 6 = 0 Factoring: x² - 3x - 2x + 6 = 0 x(x - 3) - 2(x - 3) = 0 (x - 3)(x - 2) = 0 Therefore: x = 3 or x = 2 Roots are 2 and 3.

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Completing the square method involves adding and subtracting a suitable term to make the quadratic expression a perfect square. It's used when factorization is not easily possible. We add (b/2a)² to c

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x² + 8x + 5 = 0 To complete square: x² + 8x + 16 - 16 + 5 = 0 (x + 4)² - 11 = 0 (x + 4)² = 11 x + 4 = ±√11 x = -4 ± √11 Roots: x = -4 + √11 and x = -4 - √11

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The quadratic formula is: x = (-b ± √(b² - 4ac))/2a where a ≠ 0. This gives both roots of the equation.

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Here a = 2, b = 5, c = -3 x = (-5 ± √(5² - 4(2)(-3)))/2(2) x = (-5 ± √(25 + 24))/4 x = (-5 ± √49)/4 x = (-5 ± 7)/4 x = 2/4 = 1/2 or x = -12/4 = -3 Roots: x = 1/2 and x = -3